One of the goals of electricity metering is to accurately measure the use or consumption of electrical energy resources. With such measurements, the cost of generating and delivering electricity may be allocated among consumers in relatively logical manner. Another goal of electricity metering is help identify electrical energy generation and delivery needs. For example, cumulative electricity consumption measurements for a service area can help determine the appropriate sizing of transformers and other equipment.
Electricity metering often involves the measurement of consumed power or energy in the form of watts or watt-hours. Watt-hour measurements relate directly to the actual energy that the load requires. However, the amount of watts supplied to a load does not necessarily reflect the amount of energy that must be produced by the source. In particular, the amount of load watts, or load watt-hours, does not necessarily accurately relate to the size of the service (transformers size, size of power lines, etc.) needed to supply the load. This is due in part to loads that have capacitive of inductive components. In such loads, the actual energy consumption in watt-hours is significantly less than the volt-amp-hours (VA-hours) that must be produced by the energy source.
For example, consider two loads: a first load consuming 240 watts at 120 volts and which is completely resistive, and a second load consuming 240 watts at 120 volts and having a phase difference between voltage and current of 30°. Using the basic AC power consumption equationWatts=VrmsIrms cos θ,it can be seen that the first load requires 2 amps of current because I=240/(120*cos 0°), while the second load requires 2.31 amps of current because I=240/(120*cos 30°). While the actual watt-hour consumption of the second load is the same as the first load, the second load requires more current, which can affect sizing of power lines, transformers, etc. Consequently, it can be desirable to measure VA or VA-hours to determining the size of the source, i.e. transformers size, size of the power lines, etc. needed to supply the load.
To this end, in a case of a customer that consumes significantly more VA than watts, the metering of only watt-hours results in a case in which the revenue from the customer does not directly cover the customer's proportional cost of the power delivery equipment. For this reason a more complex rate structure involving VA or VA-hours is often used to recover the investment costs for such items as transformers and power lines etc. providing energy to the load. As a consequence, many electricity meters, particularly for larger non-residential loads, have at least some capability to measure VA or VA-hours.
The calculation of VA or VA-hours in single phase systems is relatively straight forward when the signals are pure sine waves. However, if harmonics are present in the power line signal, then the calculations of VA and the practical significance of the calculated VA becomes more complex.
One common method of calculating VA involves multiplying the RMS voltage by the RMS current, or in other words VA=VRMS*IRMS. Converting VA to VA-hours, as is well known in the art, merely involves integrating the VA values over time. For example, the VA value may be calculated at ⅓rd second intervals, with each calculation considered to be the VA consumption over that ⅓rd second, or 1/10,800 of an hour. These values are then accumulated to provide a running meter of consumed VA-hours. As such calculations are routine, the terms VA and VA-hours may be used somewhat interchangeably herein, with the understanding that VA-hours may always be calculated from VA values.
In any event, a second common method of calculating VA involves first determining the value of the reactive VA, also known as VAR (Volt Amp Reactive), and actual power in watts. The method then involves deriving VA using the formula VA=√{square root over (Watt2+VAR2)}. The VAR value may be calculated using the equation VAR=VRMS*IRMS*sin θ, or by sampling voltage and current and multiplying samples of voltage and current that are 90° phase separated. If harmonics are present in the power line signal, then the use of the formula VA=√{square root over (Watt2+VAR2)} to calculate VA will yield a result that is less than that calculated from the RMS values of voltage and current, VA=VRMS*IRMS. Because of this inaccuracy, sometimes a 3rd quantity, distortion power (DP) is sometimes added as follows: VA=√{square root over (Watt2+VAR2+DP2)}.
The above equations relate generally to single phase systems. In a polyphase system, the calculation of VA is more complex and the practical significance of what is calculated goes beyond that of single phase systems. In particular, the two methods of calculating VA (or VA-hours) described above for single phase systems do not necessarily yield the same results if applied to polyphase systems even under conditions of pure sine wave signals.
In one method, VA is calculated from the RMS values of the individual phase voltages and currents for each of a polyphase system, and then the VA value for the different phases is totaled. In other words, the RMS VA of each phase is determined using VA=VRMS*IRMS and then the total VA is calculated by simply adding the individual VA of each phase. This method of calculating VA is sometimes referred to as “RMS VA” (VARMS) or “arithmetic VA”.
In another method, the VA is calculated using watts and VAR. In this method, the total amount of watts for all three phases is determined, and the amount of VAR for all three phases is determined. The total VA is then calculated using the formula VA=√{square root over (Watt2+VAR2)} where Watt and VAR represent the total load watt and VAR respectively. This method of calculating VA is sometimes referred to as “vector VA” (VAV). In this vector VA calculation, it is possible for the watts of any given phase to be negative and therefore subtracted from that of the other phases. This makes it possible for the total load watts to be less then the sum of the absolute value of the individual phase watts. Similarly it is possible for the total VAR value to be smaller then the sum of the absolute value of the individual phase VAR values.
The RMS method of calculating VA (i.e. arithmetic VA) is directly impacted by harmonics since the RMS value of a signal is directly impacted by harmonics. In contrast, the vector VA method of calculating VA tends to be minimally impacted by harmonics. For example, if there were only harmonics present in the current waveform and not in the voltage waveform, then the RMS value of the current would be larger than that of only the fundamental. Since the arithmetic VA (VArms) is the product of RMS voltage times RMS current, VArms would be larger than that calculated from just the fundamental. In contrast with no harmonics present in the voltage signal the value of watts and VARs would be unaffected by the harmonics in the current signal. Consequently, the vector VA value tends to be minimally impacted by harmonics.
It is possible to derive an arithmetic VA value using watt and VAR information. If, in the formula VA=√{square root over (Watt2+VAR2)}, the sum of the absolute value of watts from each phase is used and similarly the sum of the absolute value of VAR from each phase is used, then the resultant VA would be equivalent to arithmetic VA for pure sine wave signals. If it is desirable to compare arithmetic VA to vector VA with a similar influence of harmonics on both VA values, then this alternate method of calculating arithmetic VA may be desirable.
Accordingly, while metering VA or VA-hours can provide (or contribute to) a more accurate measure of the cost providing energy to a customer, there are various methods for calculating VA that provide varying measurements.